Crossover from quantum to Boltzmann transport in graphene

We compare a fully quantum-mechanical numerical calculation of the conductivity of graphene to the semiclassical Boltzmann theory. Considering a disorder potential that is smooth on the scale of the lattice spacing, we find quantitative agreement between the two approaches away from the Dirac point. At the Dirac point the two theories are incompatible at weak disorder, although they may be compatible for strong disorder. Our numerical calculations provide a quantitative description of the full crossover between the quantum and semiclassical graphene transport regimes.

Figure 1

(Color online) Resistance $R=1/G$ (left) and conductivity $\sigma$ obtained using Eq. (4) (right), as a function of sample length $L$. The three curves shown are for $W/\xi =200$, $K_0=2$, and $\pi n{\xi }^2=0$, 0.25, and 1 [from top to bottom (bottom to top) in left (right) panel]. Dashed lines in the right panel show $\text{d}\sigma /d\text{ln}L=4e^2/\pi h$. The inset in the left panel shows the crossover to diffusive transport $\left(L⪢\xi \right)$.

Figure 2

(Color online) Conductivity $\sigma$ versus carrier density $n$. The left panel shows the low-density behavior near the Dirac point; the right panel shows the high-density behavior. Data points are from numerical simulation with $K_0=1$ (squares) and $K_0=4$ (diamonds) with dotted lines in the left panel as a guide to the eyes. The dashed curves show the predictions of the Boltzmann theory, and the solid lines show the self-consistent Boltzmann result.

Figure 3

(Color online) Conductivity $\sigma$ versus disorder strength at the Dirac point (left) and at carrier density $\pi n=K_0/{\left(\pi \xi \right)}^2$, corresponding to the edge of the minimum conductivity plateau of Ref. 8 (right). Data points are from the numerical calculation for $L=50\xi$, and the (solid) dashed curves represent the (self-consistent) Boltzmann theory.

Figure 4

(Color online) Same as Fig. 3 but for ${\sigma }^{\prime }={\text{lim}}_{L\to \infty }\left[\sigma \left(L\right)-{\pi }^{-1}\text{ln}\left(L/\xi \right)\right]$.